I will review the published quasar absorption line constraints on variations in the fine-structure constant, alpha, focusing on the apparent disagreement between those derived from Keck/HIRES and VLT/UVES spectra which have provided evidence for and against alpha variation, respectively. I demonstrate simple yet fundamental flaws in the UVES constraints which preclude reliable comparison with those from HIRES. I will outline our program to obtain a definitive UVES measurement. I will also present several new absorption line constraints on variations in the proton-to-electron mass ratio, mu. For the two molecular hydrogen absorbers from which previous authors found tentative evidence for mu-variation, we find robustly null results. A further two molecular hydrogen absorbers, including an entirely new system, also yield tight, null constraints. Finally, I present new, detailed comparison of a radio absorption system containing ammonia inversion and molecular rotational transitions which yields the strongest current astrophysical constraint on mu-variation, dmu/mu=[mu(z)-mu(lab)]/mu(lab)=[+0.74+/-0.47(stat)+/-0.76(sys)]x10^-6, at redshift z=0.685.